Gradient-Based Curvature-Only Time-Derivative Coupling for Surface Hopping and Coherent Switching with Decay of Mixing: Application to SO2 and Molecular Tully Models.

Time-derivative couplings (TDCs) are central ingredients in mixed quantum-classical simulations of nonadiabatic dynamics, yet their evaluation remains a key practical challenge when analytic nonadiabatic coupling vectors or wave function overlaps are unavailable. Curvature-driven approximations provide an appealing alternative by evaluating TDCs from adiabatic potential energies or gradients along nuclear trajectories, motivated by the one-dimensional Baeck-An model. In this work, we examine curvature-based TDC approximations within trajectory surface hopping (TSH) and coherent switching with decay of mixing (CSDM), and introduce a modified gradient-based curvature-only formulation that retains the directional-curvature contribution to the TDC while omitting the explicit acceleration-dependent term at all times in the coupling evaluation. The performance of energy-based, gradient-based, and curvature-only TDCs is evaluated for four benchmark systems of increasing dynamical complexity: ethylene, DMABN, fulvene, and SO2. For ethylene, all curvature-driven schemes closely reproduce reference dynamics obtained with analytic nonadiabatic couplings. For DMABN and fulvene, which involve repeated nonadiabatic passages and back-transfer, the curvature-only approximation yields improved agreement with reference populations by reducing overpopulation effects observed with standard curvature formulations. In the multistate singlet-triplet dynamics of SO2, standard curvature-based TDCs introduce spurious population transfer to the ground state, an artifact that is significantly decreased by the curvature-only scheme in both TSH and CSDM. Overall, the results show that removing the acceleration-dependent contribution leads to a more robust curvature-driven TDC approximation across a wide range of nonadiabatic regimes, while retaining the conceptual simplicity and efficiency of curvature-based approximations.